The \(1+1\) -dimensional Korteweg-de Vries (KdV) equation [1]} is applicable to many real-life phenomena such as flow of a shallow fluid. Being third order in space derivative, though this non-linear partial differential equation (PDE) has a dispersion odd in momentum powers, it is completely integrable [2]} and supports localized soliton solutions [3]} that can represent different observable physical objects in fluid dynamics like tidal waves. Solitonic structures are well-known in other nonlinear systems in \(1+1\) -dimensions such as the nonlinear Schrödinger (NLS) equation which is also integrable and may seem to be more closely related to physical phenomena as the NLS solitons have been observed in Bose-Einstein condensates, cold atoms and optics [4]}. The NLS with quadratic dispersion is more suitable for a `physical visualization' is fundamentally different from the KdV system. The KdV is geometrically connected to diffeomorphism group [5]} whereas NLS is tagged with loop algebra [6]}. Moreover the usual Lax representation [7]} of KdV system involves second and third order monic differential operators (\(L,A\) ), whereas that of the NLS system is given by \(2\times 2\)  matrices [2]}. However, it is known that in a suitable weak-coupling limit, their respective solutions map into each-other [9]}, [10]}, including their soliton solutions.
